The denominator of a fraction is 1 more than double the numerator. On adding 2…
2019
The denominator of a fraction is 1 more than double the numerator. On adding 2 to the numerator and subtracting 3 from the denominator, we obtain 1. Find the original fraction.
- A.
4/9
- B.
2/5
- C.
1/9
- D.
1/3
Attempted by 2 students.
Show answer & explanation
Correct answer: A
Concept
A fraction can be represented as n/d, where n is the numerator and d is the denominator.
A verbal relation such as “one more than double” becomes an algebraic equation, and each stated change must be applied to the numerator and denominator before solving.
Application
Let the numerator be n. The denominator is then d = 2n + 1.
After adding 2 to the numerator and subtracting 3 from the denominator, the fraction becomes (n + 2)/(d − 3) = 1.
Substitute d = 2n + 1: (n + 2)/(2n − 2) = 1.
Multiply by 2n − 2: n + 2 = 2n − 2.
Subtract n from both sides: 2 = n − 2.
Add 2 to both sides: n = 4.
Therefore, d = 2(4) + 1 = 9, and the original fraction is 4/9.
Cross-check
For 4/9, the denominator relation holds because 9 = 2 × 4 + 1. After the stated changes, the value is (4 + 2)/(9 − 3) = 6/6 = 1, confirming both conditions.